7.3: Sum and Difference Identities

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Learning Objectives

In this section, you will:

Use sum and difference formulas for cosine.
Use sum and difference formulas for sine.
Use sum and difference formulas for tangent.
Use sum and difference formulas for cofunctions.
Use sum and difference formulas to verify identities.

Figure 1 Denali (formerly Mount McKinley), in Denali National Park, Alaska, rises 20,237 feet (6,168 m) above sea level. It is the highest peak in North America. (credit: Daniel A. Leifheit, Flickr)

How can the height of a mountain be measured? What about the distance from Earth to the sun? Like many seemingly impossible problems, we rely on mathematical formulas to find the answers. The trigonometric identities, commonly used in mathematical proofs, have had real-world applications for centuries, including their use in calculating long distances.

The trigonometric identities we will examine in this section can be traced to a Persian astronomer who lived around 950 AD, but the ancient Greeks discovered these same formulas much earlier and stated them in terms of chords. These are special equations or postulates, true for all values input to the equations, and with innumerable applications.

In this section, we will learn techniques that will enable us to solve problems such as the ones presented above. The formulas that follow will simplify many trigonometric expressions and equations. Keep in mind that, throughout this section, the term formula is used synonymously with the word identity.

Using the Sum and Difference Formulas for Cosine

Finding the exact value of the sine, cosine, or tangent of an angle is often easier if we can rewrite the given angle in terms of two angles that have known trigonometric values. We can use the special angles, which we can review in the unit circle shown in Figure 2.

Diagram of the unit circle with points labeled on its edge. P point is at an angle a from the positive x axis with coordinates (cosa, sina). Point Q is at an angle of B from the positive x axis with coordinates (cosb, sinb). Angle POQ is a - B degrees. Point A is at an angle of (a-B) from the x axis with coordinates (cos(a-B), sin(a-B)). Point B is just at point (1,0). Angle AOB is also a - B degrees. Radii PO, AO, QO, and BO are all 1 unit long and are the legs of triangles POQ and AOB. Triangle POQ is a rotation of triangle AOB, so the distance from P to Q is the same as the distance from A to B.

Figure 2 The Unit Circle

We will begin with the sum and difference formulas for cosine, so that we can find the cosine of a given angle if we can break it up into the sum or difference of two of the special angles. See Table 1.

Sum formula for cosine	cos(α+β)=cosαcosβ−sinαsinβcos(α+β)=cosαcosβ−sinαsinβ
Difference formula for cosine	cos(α−β)=cosαcosβ+sinαsinβcos(α−β)=cosαcosβ+sinαsinβ

Table 1

First, we will prove the difference formula for cosines. Let’s consider two points on the unit circle. See Figure 3. Point PP is at an angle αα from the positive x-axis with coordinates (cosα,sinα)(cosα,sinα) and point QQ is at an angle of ββ from the positive x-axis with coordinates (cosβ,sinβ).(cosβ,sinβ). Note the measure of angle POQPOQ is α−β.α−β.

Label two more points: AA at an angle of (α−β)(α−β) from the positive x-axis with coordinates (cos(α−β),sin(α−β));(cos(α−β),sin(α−β)); and point BB with coordinates (1,0).(1,0). Triangle POQPOQ is a rotation of triangle AOBAOB and thus the distance from PP to QQ is the same as the distance from AA to B.B.

Figure 3

We can find the distance from PP to QQ using the distance formula.

dPQ=(cosα−cosβ)2+(sinα−sinβ)2−−−−−−−−−−−−−−−−−−−−−−−−−−−−√ =cos2α−2cosαcosβ+cos2β+sin2α−2sinαsinβ+sin2β−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−√dPQ=(cosα−cosβ)2+(sinα−sinβ)2 =cos2α−2cosαcosβ+cos2β+sin2α−2sinαsinβ+sin2β

Then we apply the Pythagorean Identity and simplify.

=(cos2α+sin2α)+(cos2β+sin2β)−2cosαcosβ−2sinαsinβ−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−√=1+1−2cosαcosβ−2sinαsinβ−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−√=2−2cosαcosβ−2sinαsinβ−−−−−−−−−−−−−−−−−−−−−−−−−−−−−√=(cos2α+sin2α)+(cos2β+sin2β)−2cosαcosβ−2sinαsinβ=1+1−2cosαcosβ−2sinαsinβ=2−2cosαcosβ−2sinαsinβ

Similarly, using the distance formula we can find the distance from AA to B.B.

dAB=(cos(α−β)−1)2+(sin(α−β)−0)2−−−−−−−−−−−−−−−−−−−−−−−−−−−−−√ =cos2(α−β)−2cos(α−β)+1+sin2(α−β)−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−√dAB=(cos(α−β)−1)2+(sin(α−β)−0)2 =cos2(α−β)−2cos(α−β)+1+sin2(α−β)

Applying the Pythagorean Identity and simplifying we get:

=(cos2(α−β)+sin2(α−β))−2cos(α−β)+1−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−√=1−2cos(α−β)+1−−−−−−−−−−−−−−−−−√=2−2cos(α−β)−−−−−−−−−−−−−−√=(cos2(α−β)+sin2(α−β))−2cos(α−β)+1=1−2cos(α−β)+1=2−2cos(α−β)

Because the two distances are the same, we set them equal to each other and simplify.

2−2cosαcosβ−2sinαsinβ−−−−−−−−−−−−−−−−−−−−−−−−−−−−−√=2−2cos(α−β)−−−−−−−−−−−−−−√ 2−2cosαcosβ−2sinαsinβ=2−2cos(α−β) 2−2cosαcosβ−2sinαsinβ=2−2cos(α−β) 2−2cosαcosβ−2sinαsinβ=2−2cos(α−β)

Finally we subtract 22 from both sides and divide both sides by −2.−2.

cosαcosβ+sinαsinβ=cos(α−β) cosαcosβ+sinαsinβ=cos(α−β)

Thus, we have the difference formula for cosine. We can use similar methods to derive the cosine of the sum of two angles.

SUM AND DIFFERENCE FORMULAS FOR COSINE

These formulas can be used to calculate the cosine of sums and differences of angles.

cos(α+β)=cosαcosβ−sinαsinβcos(α+β)=cosαcosβ−sinαsinβ

cos(α−β)=cosαcosβ+sinαsinβcos(α−β)=cosαcosβ+sinαsinβ

HOW TO

Given two angles, find the cosine of the difference between the angles.

Write the difference formula for cosine.
Substitute the values of the given angles into the formula.
Simplify.

EXAMPLE 1

Finding the Exact Value Using the Formula for the Cosine of the Difference of Two Angles

Using the formula for the cosine of the difference of two angles, find the exact value of cos(5π4−π6).cos(5π4−π6).

Answer

TRY IT #1

Find the exact value of cos(π3−π4).cos(π3−π4).

EXAMPLE 2

Finding the Exact Value Using the Formula for the Sum of Two Angles for Cosine

Find the exact value of cos(75∘).cos(75∘).

Answer

TRY IT #2

Find the exact value of cos(105∘).cos(105∘).

Using the Sum and Difference Formulas for Sine

The sum and difference formulas for sine can be derived in the same manner as those for cosine, and they resemble the cosine formulas.

SUM AND DIFFERENCE FORMULAS FOR SINE

These formulas can be used to calculate the sines of sums and differences of angles.

sin(α+β)=sinαcosβ+cosαsinβsin(α+β)=sinαcosβ+cosαsinβ

sin(α−β)=sinαcosβ−cosαsinβsin(α−β)=sinαcosβ−cosαsinβ

HOW TO

Given two angles, find the sine of the difference between the angles.

Write the difference formula for sine.
Substitute the given angles into the formula.
Simplify.

EXAMPLE 3

Using Sum and Difference Identities to Evaluate the Difference of Angles

Use the sum and difference identities to evaluate the difference of the angles and show that part a equals part b.

ⓐ sin(45∘−30∘)sin(45∘−30∘)
ⓑ sin(135∘−120∘)sin(135∘−120∘)

Answer

EXAMPLE 4

Finding the Exact Value of an Expression Involving an Inverse Trigonometric Function

Find the exact value of

sin(cos−112+sin−135).sin(cos−112+sin−135).

Answer

Using the Sum and Difference Formulas for Tangent

Finding exact values for the tangent of the sum or difference of two angles is a little more complicated, but again, it is a matter of recognizing the pattern.

Finding the sum of two angles formula for tangent involves taking quotient of the sum formulas for sine and cosine and simplifying. Recall, tanx=sinxcosx,cosx≠0.tanx=sinxcosx,cosx≠0.

Let’s derive the sum formula for tangent.

tan(α+β)=sin(α+β)cos(α+β) =sinαcosβ+cosαsinβcosαcosβ−sinαsinβ =sinαcosβ+cosαsinβcosαcosβcosαcosβ−sinαsinβcosαcosβ =sinαcosβcosαcosβ+cosαsinβcosαcosβcosαcosβcosαcosβ−sinαsinβcosαcosβ =sinαcosα+sinβcosβ1−sinαsinβcosαcosβ =tanα+tanβ1−tanαtanβDivide the numerator and denominator by cosαcosβtan(α+β)=sin(α+β)cos(α+β) =sinαcosβ+cosαsinβcosαcosβ−sinαsinβ =sinαcosβ+cosαsinβcosαcosβcosαcosβ−sinαsinβcosαcosβ Divide the numerator and denominator by cosαcosβ =sinαcosβcosαcosβ+cosαsinβcosαcosβcosαcosβcosαcosβ−sinαsinβcosαcosβ =sinαcosα+sinβcosβ1−sinαsinβcosαcosβ =tanα+tanβ1−tanαtanβ

We can derive the difference formula for tangent in a similar way.

SUM AND DIFFERENCE FORMULAS FOR TANGENT

The sum and difference formulas for tangent are:

tan(α+β)=tanα+tanβ1−tanαtanβtan(α+β)=tanα+tanβ1−tanαtanβ

tan(α−β)=tanα−tanβ1+tanαtanβtan(α−β)=tanα−tanβ1+tanαtanβ

HOW TO

Given two angles, find the tangent of the sum of the angles.

Write the sum formula for tangent.
Substitute the given angles into the formula.
Simplify.

EXAMPLE 5

Finding the Exact Value of an Expression Involving Tangent

Find the exact value of tan(π6+π4).tan(π6+π4).

Answer

TRY IT #3

Find the exact value of tan(2π3+π4).tan(2π3+π4).

EXAMPLE 6

Finding Multiple Sums and Differences of Angles

Given sinα=35,0<α<π2,cosβ=−513,π<β<3π2,sinα=35,0<α<π2,cosβ=−513,π<β<3π2, find

ⓐ sin(α+β)sin(α+β)
ⓑ cos(α+β)cos(α+β)
ⓒ tan(α+β)tan(α+β)
ⓓ tan(α−β)tan(α−β)

Answer

$Diagram of a triangle in the x,y plane. The vertices are at the origin, (4,0), and (4,3). The angle at the origin is alpha degrees, The angle formed by the x-axis and the side from (4,3) to (4,0) is a right angle. The side opposite the right angle has length 5.$

$Diagram of a triangle in the x,y plane. The vertices are at the origin, (-5,0), and (-5, -12). The angle at the origin is Beta degrees. The angle formed by the x axis and the side from (-5, -12) to (-5,0) is a right angle. The side opposite the right angle has length 13.$

Analysis

A common mistake when addressing problems such as this one is that we may be tempted to think that αα and ββ are angles in the same triangle, which of course, they are not. Also note that

tan(α+β)=sin(α+β)cos(α+β)tan(α+β)=sin(α+β)cos(α+β)

Using Sum and Difference Formulas for Cofunctions

Now that we can find the sine, cosine, and tangent functions for the sums and differences of angles, we can use them to do the same for their cofunctions. You may recall from Right Triangle Trigonometry that, if the sum of two positive angles is π2,π2, those two angles are complements, and the sum of the two acute angles in a right triangle is π2,π2, so they are also complements. In Figure 6, notice that if one of the acute angles is labeled as θ,θ, then the other acute angle must be labeled (π2−θ).(π2−θ).

Notice also that sinθ=cos(π2−θ):sinθ=cos(π2−θ): opposite over hypotenuse. Thus, when two angles are complementary, we can say that the sine of θθ equals the cofunction of the complement of θ.θ. Similarly, tangent and cotangent are cofunctions, and secant and cosecant are cofunctions.

Image of a right triangle. The remaining angles are labeled theta and pi/2 - theta.

Figure 6

From these relationships, the cofunction identities are formed.

COFUNCTION IDENTITIES

The cofunction identities are summarized in Table 2.

sinθ=cos(π2−θ)sinθ=cos(π2−θ)	cosθ=sin(π2−θ)cosθ=sin(π2−θ)
tanθ=cot(π2−θ)tanθ=cot(π2−θ)	cotθ=tan(π2−θ)cotθ=tan(π2−θ)
secθ=csc(π2−θ)secθ=csc(π2−θ)	cscθ=sec(π2−θ)cscθ=sec(π2−θ)

Table 2

Notice that the formulas in the table may also justified algebraically using the sum and difference formulas. For example, using

cos(α−β)=cosαcosβ+sinαsinβ,cos(α−β)=cosαcosβ+sinαsinβ,

we can write

cos(π2−θ)=cosπ2cosθ+sinπ2sinθ =(0)cosθ+(1)sinθ =sinθcos(π2−θ)=cosπ2cosθ+sinπ2sinθ =(0)cosθ+(1)sinθ =sinθ

EXAMPLE 7

Finding a Cofunction with the Same Value as the Given Expression

Write tanπ9tanπ9 in terms of its cofunction.

Answer

TRY IT #4

Write sinπ7sinπ7 in terms of its cofunction.

Using the Sum and Difference Formulas to Verify Identities

Verifying an identity means demonstrating that the equation holds for all values of the variable. It helps to be very familiar with the identities or to have a list of them accessible while working the problems. Reviewing the general rules from Solving Trigonometric Equations with Identities may help simplify the process of verifying an identity.

HOW TO

Given an identity, verify using sum and difference formulas.

Begin with the expression on the side of the equal sign that appears most complex. Rewrite that expression until it matches the other side of the equal sign. Occasionally, we might have to alter both sides, but working on only one side is the most efficient.
Look for opportunities to use the sum and difference formulas.
Rewrite sums or differences of quotients as single quotients.
If the process becomes cumbersome, rewrite the expression in terms of sines and cosines.

EXAMPLE 8

Verifying an Identity Involving Sine

Verify the identity sin(α+β)+sin(α−β)=2sinαcosβ.sin(α+β)+sin(α−β)=2sinαcosβ.

Answer

EXAMPLE 9

Verifying an Identity Involving Tangent

Verify the following identity.

sin(α−β)cosαcosβ=tanα−tanβsin(α−β)cosαcosβ=tanα−tanβ

Answer

TRY IT #5

Verify the identity: tan(π−θ)=−tanθ.tan(π−θ)=−tanθ.

EXAMPLE 10

Using Sum and Difference Formulas to Solve an Application Problem

Let L1L1 and L2L2 denote two non-vertical intersecting lines, and let θθ denote the acute angle between L1L1 and L2.L2. See Figure 7. Show that

tanθ=m2−m11+m1m2tanθ=m2−m11+m1m2

where m1m1 and m2m2 are the slopes of L1L1 and L2L2 respectively. (Hint: Use the fact that tanθ1=m1tanθ1=m1 and tanθ2=m2.tanθ2=m2. )

Diagram of two non-vertical intersecting lines L1 and L2 also intersecting the x-axis. The acute angle formed by the intersection of L1 and L2 is theta. The acute angle formed by L2 and the x-axis is theta 1, and the acute angle formed by the x-axis and L1 is theta 2.

Figure 7

Answer

EXAMPLE 11

Investigating a Guy-wire Problem

For a climbing wall, a guy-wire RR is attached 47 feet high on a vertical pole. Added support is provided by another guy-wire SS attached 40 feet above ground on the same pole. If the wires are attached to the ground 50 feet from the pole, find the angle αα between the wires. See Figure 8.

Two right triangles. Both share the same base, 50 feet. The first has a height of 40 ft and hypotenuse S. The second has height 47 ft and hypotenuse R. The height sides of the triangles are overlapping. There is a B degree angle between R and the base, and an a degree angle between the two hypotenuses within the B degree angle.

Figure 8

Answer

Analysis

Occasionally, when an application appears that includes a right triangle, we may think that solving is a matter of applying the Pythagorean Theorem. That may be partially true, but it depends on what the problem is asking and what information is given.

MEDIA

Access these online resources for additional instruction and practice with sum and difference identities.

7.2 Section Exercises

Verbal

Explain the basis for the cofunction identities and when they apply.

Is there only one way to evaluate cos(5π4)?cos(5π4)? Explain how to set up the solution in two different ways, and then compute to make sure they give the same answer.

Explain to someone who has forgotten the even-odd properties of sinusoidal functions how the addition and subtraction formulas can determine this characteristic for f(x)=sin(x)f(x)=sin(x) and g(x)=cos(x).g(x)=cos(x). (Hint: 0−x=−x0−x=−x )

Algebraic

For the following exercises, find the exact value.

cos(7π12)cos(7π12)

cos(π12)cos(π12)

sin(5π12)sin(5π12)

sin(11π12)sin(11π12)

tan(−π12)tan(−π12)

tan(19π12)tan(19π12)

For the following exercises, rewrite in terms of sinxsinx and cosx.cosx.

10.

sin(x+11π6)sin(x+11π6)

11.

sin(x−3π4)sin(x−3π4)

12.

cos(x−5π6)cos(x−5π6)

13.

cos(x+2π3)cos(x+2π3)

For the following exercises, simplify the given expression.

14.

csc(π2−t)csc(π2−t)

15.

sec(π2−θ)sec(π2−θ)

16.

cot(π2−x)cot(π2−x)

17.

tan(π2−x)tan(π2−x)

18.

sin(2x)cos(5x)−sin(5x)cos(2x)sin(2x)cos(5x)−sin(5x)cos(2x)

19.

tan(32x)−tan(75x)1+tan(32x)tan(75x)tan(32x)−tan(75x)1+tan(32x)tan(75x)

For the following exercises, find the requested information.

20.

Given that sina=23sina=23 and cosb=−14,cosb=−14, with aa and bb both in the interval [π2,π),[ π2,π ), find sin(a+b)sin(a+b) and cos(a−b).cos(a−b).

21.

Given that sina=45,sina=45, and cosb=13,cosb=13, with aa and bb both in the interval [0,π2),[ 0,π2 ), find sin(a−b)sin(a−b) and cos(a+b).cos(a+b).

For the following exercises, find the exact value of each expression.

22.

sin(cos−1(0)−cos−1(12))sin(cos−1(0)−cos−1(12))

23.

cos(cos−1(2√2)+sin−1(3√2))cos(cos−1(22)+sin−1(32))

24.

tan(sin−1(12)−cos−1(12))tan(sin−1(12)−cos−1(12))

Graphical

For the following exercises, simplify the expression, and then graph both expressions as functions to verify the graphs are identical.

25.

cos(π2−x)cos(π2−x)

26.

sin(π−x)sin(π−x)

27.

tan(π3+x)tan(π3+x)

28.

sin(π3+x)sin(π3+x)

29.

tan(π4−x)tan(π4−x)

30.

cos(7π6+x)cos(7π6+x)

31.

sin(π4+x)sin(π4+x)

32.

cos(5π4+x)cos(5π4+x)

For the following exercises, use a graph to determine whether the functions are the same or different. If they are the same, show why. If they are different, replace the second function with one that is identical to the first. (Hint: think 2x=x+x.2x=x+x. )

33.

f(x)=sin(4x)−sin(3x)cosxf(x)=sin(4x)−sin(3x)cosx, g(x)=sinxcos(3x)g(x)=sinxcos(3x)

34.

f(x)=cos(4x)+sinxsin(3x),g(x)=−cosxcos(3x)f(x)=cos(4x)+sinxsin(3x),g(x)=−cosxcos(3x)

35.

f(x)=sin(3x)cos(6x)f(x)=sin(3x)cos(6x), g(x)=−sin(3x)cos(6x)g(x)=−sin(3x)cos(6x)

36.

f(x)=sin(4x)f(x)=sin(4x), g(x)=sin(5x)cosx−cos(5x)sinxg(x)=sin(5x)cosx−cos(5x)sinx

37.

f(x)=sin(2x)f(x)=sin(2x), g(x)=2sinxcosxg(x)=2sinxcosx

38.

f(θ)=cos(2θ)f(θ)=cos(2θ), g(θ)=cos2θ−sin2θg(θ)=cos2θ−sin2θ